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The given quadratic function is:

\[ y = \frac{1}{2}x^2 + 3x - \frac{5}{2} \]

We need to determine whether the function has a maximum or minimum value and find that value. Additionally, we need to find the value of \( x \) at which this maximum or minimum occurs.

### Step-by-Step Solution:

1. **Identify the form of the quadratic function**:
   A quadratic function is generally given by:
   \[ y = ax^2 + bx + c \]
   where \( a = \frac{1}{2} \), \( b = 3 \), and \( c = -\frac{5}{2} \).

2. **Determine the direction of the parabola**:
   The coefficient \( a = \frac{1}{2} \) is positive, indicating that the parabola opens upwards. Therefore, the function has a minimum value.

3. **Find the vertex of the parabola**:
   The vertex form of a quadratic function gives the minimum (or maximum) point. The x-coordinate of the vertex is found using:
   \[ x = -\frac{b}{2a} \]
   Substituting \( a \) and \( b \):
   \[ x = -\frac{3}{2 \times \frac{1}{2}} = -\frac{3}{1} = -3 \]

4. **Calculate the minimum value**:
   Substitute \( x = -3 \) back into the original function to find \( y \):
   \[ y = \frac{1}{2}(-3)^2 + 3(-3) - \frac{5}{2} \]
   \[ y = \frac{1}{2}(9) + (-9) - \frac{5}{2} \]
   \[ y = \frac{9}{2} - 9 - \frac{5}{2} \]
   \[ y = \frac{9}{2} - \frac{18}{2} - \frac{5}{2} \]
   \[ y = \frac{9 - 18 - 5}{2} \]
   \[ y = \frac{-14}{2} \]
   \[ y = -7 \]

### Conclusion:
- The function \( y = \frac{1}{2}x^2 + 3x - \frac{5}{2} \) has a minimum value.
- The minimum value is \( -7 \).
- This occurs when \( x = -3 \).

### Completing the Given Statements:
函数y有最小 (选填1: 大; 2: 小) 值, 值 = -7. 当x = -3时，随x的增大而减小。

Would you like any further details or explanations? 

Here are 5 questions you can ask next:
1. How do you find the vertex of a quadratic function in standard form?
2. Can you explain how to convert a quadratic function to vertex form?
3. What are the properties of parabolas in quadratic functions?
4. How do you determine the axis of symmetry for a quadratic function?
5. Can you explain the significance of the discriminant in quadratic equations?

**Tip**: When dealing with quadratic functions, always check the sign of the leading coefficient to determine if the parabola opens upwards (minimum value) or downwards (maximum value).

Learn how to determine the minimum value of the quadratic function y = (1/2)x^2 + 3x - (5/2) with a step-by-step solution. This problem covers key concepts including quadratic functions, finding the vertex of a parabola, and identifying minimum and maximum values. Suitable for students in Grades 10-12.

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